Sur quelques extensions au cadre Banachique de la notion d'op\'erateur de Hilbert-Schmidt

TL;DR

This paper explores various extensions of Hilbert-Schmidt operators to Banach spaces, introducing new classes like pre-Hilbert-Schmidt operators and analyzing their properties and relationships.

Contribution

It defines the class of pre-Hilbert-Schmidt operators in Banach spaces and investigates their properties, extending the classical Hilbert-Schmidt concept beyond Hilbert spaces.

Findings

Introduction of the class PS_2(E; F) of pre-Hilbert-Schmidt operators.

Characterization of these operators in the context of Banach spaces.

Connection to existing classes like p-summing and γ-radonifying operators.

Abstract

In this work we discuss several ways to extend to the context of Banach spaces the notion of Hilbert-Schmidt operators: $p$-summing operators, $\gamma$-summing or $\gamma$-radonifying operators, weakly $*1$-nuclear operators and classes of operators defined via factorization properties. We introduce the class $PS_2(E; F)$ of pre-Hilbert-Schmidt operators as the class of all operators $u:E\to F$ such that $w\circ u \circ v$ is Hilbert-Schmidt for every bounded operator $v: H_1\to E$ and every bounded operator $w:F\to H_2$, where $H_1$ et $H_2$ are Hilbert spaces. Besides the trivial case where one of the spaces $E$ or $F$ is a "Hilbert-Schmidt space", this space seems to have been described only in the easy situation where one of the spaces $E$ or $F$ is a Hilbert space.